A two-variable Iwasawa main conjecture over the eigencurve
David Hansen
January 20, 2015
1
The setup
∼
Fix an odd prime p, an algebraic closure Qp , and an isomorphism C → Qp . Fix an integer
N ≥ 1 prime to p, and let T be the polynomial algebra over Z generated by the operators
rig be the rigid
T! , ! ! N p, Up and #d$ , d ∈ (Z/N Z)× . Set W = Spf(Zp [[Z×
p ]]), and let W = W
×
×
analytic space of characters of Zp together with its universal character χW : Z×
p → O(W ) ;
we embed Z in W (Qp ) by mapping k to the character t &→ tk−2 . For any λ ∈ W (Qp ) we
(slightly abusively) write
Mλ† (Γ1 (N )) ⊂ Qp [[q]]
for the space of q-expansions of overconvergent modular forms of weight λ and tame level N .
Let C (N ) be the tame level N eigencurve, with weight map w : C (N ) → W and universal
Hecke algebra homomorphism φ : T → O(C (N )).
Let C0M −new (N ), for M |N , be the Zariski closure of the points associated with classical
cuspidal newforms of level Γ1 (M ); this sits inside C (N ) as a union of irreducible components.
Finally, let C = CN be the normalization of C0N −new (N ), with its natural morphism i :
CN → C (N ). This is a disjoint union of smooth, reduced rigid analytic curves. We slightly
abusively write w = w ◦ i, φ = i∗ φ, etc.
rig
×
Let X = Spf(Zp [[Z×
p ]]) . A point x ∈ X (Qp ) defines a continuous character ψx : Zp →
!
×
Qp . This space will be our “cyclotomic variable”. There is a partition X = X + X − ,
where x ∈ X ± according to ψx (−1) = ±1. Set YN = CN × X and YN± = CN × X ± ⊂ Y .
Let pr and pr± denote the projections of YN and YN± onto CN , respectively.
Our goal is to define the following objects:
• Torsion-free coherent sheaves V ± (N ) on C (N ). In the notation introduced below, we
have
H 0 (CΩ,h (N ), V ± (N )) ∼
= SymbΓ1 (N p) (DΩ )±
h.
On C0N −new (N ) these sheaves have generic rank one.
1
• Torsion-free coherent sheaves
"
#
M ± (N ) = H omw−1 OW V ± (N ), w−1 OW
on C (N ). We need to be careful about the meaning of the right-hand side, since the
morphism w isn’t finite. Again, these sheaves have generic rank one on C0N −new (N ).
Set
$± = (i ◦ pr )∗ M ± (N )
M
±
N
= pr∗± (i∗ M ± (N )).
Since i∗ M ± (N ) is torsion-free of generic rank one on a smooth reduced curve, it
$± is locally free of rank one (and in fact,
is locally free of rank one. Therefore, M
N
±
$±
H 0 (pr−1
± (U ), MN ) is free of rank one over O(U × X ) for suitable affinoids U ⊂ CN ).
• A canonical global section
%
&
$N .
L ∈ H 0 YN , M
$N denotes the natural line bundle on YN which restricts to M
$± on Y ± . The
Here M
N
N
element L is the canonical two-variable p-adic L-function on CN × X , in a sense we
will make precise.
Since L is a section of a line bundle on a normal rigid analytic space, it generates a coherent
ideal sheaf IL ⊂ OY in the usual way. The general philosophy of Iwasawa theory requires
that IL coincide with the characteristic ideal of a suitable sheaf of Selmer groups over Y .
We have a candidate for this sheaf.
The eigencurve
We very briefly recall the “eigencurve of modular symbols.” The main references here are
Bellaiche’s “Critical p-adic L-functions”, Stevens’s “Rigid analytic modular symbols”, and
my eigenvarieties paper.
Let s be a nonnegative integer. Consider the ring of functions
As = {f : Zp → Qp | f analytic on each ps Zp − coset} .
(
x
Recall that by a fundamental result of Amice, the functions
=
define
j
an orthonormal basis of As . The ring As is affinoid, and we set Bs = Sp(As ), so e.g.
)
*
Bs (Cp ) = x ∈ Cp | inf |x − a| ≤ p−s .
esj (x)
a∈Zp
2
,p−s j-!
'
×
Given an affinoid open Ω ⊂ W with associated character χΩ : Z×
p → O(Ω) , define
AsΩ = O(Bs × Ω)
+
= As ⊗O(Ω).
b+dx
For suitably large s, we have the left action (γ · f )(x) = χΩ (a + cx)f ( a+cx
) of Γ0 (p) on this
s
s
module. Let DΩ be the O(Ω)-Banach dual of AΩ with dual right action, and set
DΩ =
lim DsΩ
∞←s
∼
+
= D(Zp )⊗O(Ω).
The assignment Ω ! DΩ is a “Frechet sheaf” on W . Define
MΩ† (N ) = SymbΓ1 (N )∩Γ0 (p) (DΩ ).
Here, for any neat Γ < SL2 (Z) and any right Γ-module Q,
,
SymbΓ (Q) := f ∈ HomZ (Div0 P1 (Q), Q) | f (γD) ·Q γ = f (D) ∀γ ∈ Γ, D ∈ Div0 .
'
(
−1
†
The algebra T acts naturally on MΩ (N ). The matrix
conjugates Γ1 (N ) ∩ Γ0 (p)
1
to itself, and so induces an order two automorphism ι of MΩ† (N ). We write MΩ† (N )± =
†
1±ι
2 ◦ MΩ (N ) for the two eigenspaces
. of /ι.
For h ∈ Q, let B[h] = SpQp ph X be the closed rigid ball of radius ph , and A1 =
∪h B[h]. Let F (T ) ∈ O(W ){{X}} be the Fredholm series such that
F |Ω = det(1 − Up X)|MΩ† (N )
for all Ω. This cuts out a Fredholm hypersurface Z ⊂ W × A1 . Let w be the projection
W ×A1 → W . We say (Ω, h) is a slope datum if MΩ† (N ) admits a slope-≤ h direct summand
MΩ† (N )h . This is a finite projective O(Ω)-module, and is Hecke-stable. Note that (Ω, h) is
a slope datum if and only if ZΩ,h := Z ∩ (Ω × B[h]) is slope-adapted, i.e. finite flat over
Ω and disconnected from its complement in ZΩ := Z ∩ w−1 (Ω). The set of slope-adapted
ZΩ,h ’s give an admissible covering of Z (this last is a foundational result of Buzzard). The
map “X → Up−1 ” makes MΩ† (N )h into a finite O(ZΩ,h )-module.
%
&
Let TΩ,h denote the subalgebra of EndO(Ω) MΩ† (N )h generated by the image of T ⊗Z
O(Ω). This is finite over O(Ω), so affinoid, and the map O(Ω) → TΩ,h factors through
O(Ω) → O(ZΩ,h ) → TΩ,h . The affinoids CΩ,h (N ) := SpTΩ,h → ZΩ,h glue over the covering of Z by the ZΩ,h ’s into C (N ). (And from now on we write TΩ,h and O(CΩ,h (N ))
±
interchangeably.) Each MΩ† (N )±
h is a finite TΩ,h -module, and we define V (N ) to be the
coherent sheaf on C (N ) obtained by gluing them up.
3
The construction
%
&
±
Proposition. The TΩ,h -modules MΩ,h
(N ) := HomO(Ω) MΩ† (N )±
,
O(Ω)
glue into coherh
ent sheaves M ± (N ) over C (N ).
Proof. The key point is the following: suppose ZΩ" ,h" ⊂ ZΩ,h is an inclusion of slopeadapted affinoids. We may assume Ω) ⊂ Ω and h) ≤ h, so the inclusion ZΩ" ,h" ⊂ ZΩ,h
!
"
“factors” as ZΩ" ,h" ⊂ ZΩ" ,h ⊂ ZΩ,h where ZΩ" ,h = ZΩ" ,h" ZΩ>h
" ,h is slope-adapted with both
pieces finite flat over Ω. Then TΩ,h ⊗O(Ω) O(Ω) ) ∼
= TΩ" ,h , and
%
&
%
&
†
±
∼
"
M
(N
)
,
O(Ω)
⊗TΩ,h TΩ,h ⊗O(Ω) O(Ω) )
HomO(Ω) MΩ† (N )±
,
O(Ω)
⊗
T
Hom
=
T
Ω ,h
O(Ω)
Ω,h
h
Ω
h
%
&
†
∼
,
O(Ω)
⊗O(Ω) O(Ω) )
= HomO(Ω) MΩ (N )±
h
&
%
†
)
)
∼
⊗
O(Ω
),
O(Ω
)
= HomO(Ω" ) MΩ (N )±
O(Ω)
h
%
&
†
)
∼
,
O(Ω
)
= HomO(Ω" ) MΩ" (N )±
h
as TΩ" ,h -modules. The third line here follows from flatness of O(Ω) ) over O(Ω), and
the fourth line from a basic property of slope decompositions. Now getting from TΩ" ,h modules to TΩ" ,h" -modules is easy, because everything in sight has an idempotent decomposition coming from the previously described set-theoretic decomposition of ZΩ" ,h . So
±
MΩ,h
(N ) ⊗TΩ,h TΩ" ,h" ∼
= MΩ±" ,h" (N ) as desired. "
×
Let D(Z×
p ) denote the ring of locally analytic Qp -valued distributions on Zp , and let
×
±
D(Zp ) denote the subspace of distributions for which
0
0
f (x)µ(x).
f (−x)µ(x) = ±
Z×
p
Z×
p
∼
Recall the Amice isomorphism D(Z×
p ) = O(X ); this is an isomorphism of Frechet Qp ± ∼
±
algebras, and induces isomorphisms D(Z×
p ) = O(X ).
Proposition. There is a natural isomorphism
&
%
× ± ∼
±
+
(N ) ⊗O(CΩ,h (N )) O(CΩ,h (N ) × X ± )
HomO(Ω) MΩ† (N )±
,
O(Ω)
⊗D(Z
)
= MΩ,h
p
h
compatible with all structures.
Proof. By the Amice isomorphism, so we have
%
&
%
&
†
× ± ∼
±
±
+
M
(N
)
,
O(Ω
×
X
)
.
HomO(Ω) MΩ† (N )±
,
O(Ω)
⊗D(Z
)
Hom
=
O(Ω)
p
Ω
h
h
4
Choose an increasing cover of X ± by affinoids X1± ⊂ · · · ⊂ Xn± ⊂ · · · . Then
%
&
%
&
±
±
HomO(Ω) MΩ† (N )±
= HomO(Ω) MΩ† (N )±
h , O(Ω × Xn )
h , O(Ω) ⊗O(Ω) O(Ω × Xn )
&
%
,
O(Ω)
⊗TΩ,h TΩ,h ⊗O(Ω) O(Ω × Xn± )
= HomO(Ω) MΩ† (N )±
h
%
&
±
+
,
O(Ω)
⊗TΩ,h TΩ,h ⊗O(Ω) O(Ω)⊗O(X
= HomO(Ω) MΩ† (N )±
n )
h
%
&
±
+
= HomO(Ω) MΩ† (N )±
,
O(Ω)
⊗TΩ,h TΩ,h ⊗O(X
n )
h
±
= MΩ,h
(N ) ⊗O(CΩ,h (N )) O(CΩ,h (N ) × Xn± ).
The second, third, and fifth equalities here are trivial. The first equality follows from the
identification HomR (M, N ) ⊗R P ∼
= HomR (M, N ⊗R P ) for M, N, P any R-modules with
M finitely presented and P flat, together with the flatness of O(Ω × Xn± ) over O(Ω).1 The
fourth equality evidently follows from the identity
± ∼
±
+
+
TΩ,h ⊗O(Ω) O(Ω)⊗O(X
n ) = TΩ,h ⊗O(Xn ),
which is an easy consequence of the ONability of O(Xn± ) and the module-finiteness of TΩ,h
over O(Ω) (or, more conceptually, use Propositions 3.7.3/6 and 2.1.7/7 of BGR). Passing to
the inverse limit over n, we conclude. "
Definition / Claim. The sheaf
$± (N ) := pr∗ M ± (N )
M
±
on C (N ) × X ± is characterized by the isomorphism
±
× ±
$± (N )) ∼
+
H 0 (CΩ,h (N ) × X ± , M
(N ) ⊗TΩ,h TΩ,h ⊗D(Z
= MΩ,h
p)
× ±
+
= HomO(Ω) (MΩ† (N )±
h , O(Ω)⊗D(Zp ) ).
Proof. This is an easy consequence of the previous proposition. "
$± on Y ± is the pullback of M
$± (N ) under i×id : CN ×X ± →
Definition. The sheaf M
N
N
±
C (N ) × X .
Given any Ω, let LΩ (N ) denote the composite map
MΩ† (N ) = SymbΓ1 (N )∩Γ0 (p) (DΩ )
Φ*→Φ((∞)−(0))
−→
1
+
+
DΩ ∼
→ D(Z×
= D(Zp )⊗O(Ω)
p )⊗O(Ω).
This flatness can be deduced by factoring the map as O(Ω) → O(Ω × X) → O(Ω × Xn ) where X is some
finite disjoint union of B[0]’s and Xn ⊂ X is an affinoid subdomain. The second arrow is then flat by the
flatness of coordinate rings of affinoid subdomains, and the first arrow is flat because A → A $X% is flat for
any Tate algebra A.
5
×
The final arrow here is induced by the map D(Zp ) → D(Z×
p ) dual to the map A (Zp ) →
†
A (Zp ) given by extending functions by zero. Since LΩ (N ) is O(Ω)-linear and MΩ (N )h is
an O(Ω)-module direct summand of MΩ† (N ) (and likewise the ±-subspaces), we may regard
the restriction L±
Ω,h (N ) := LΩ (N )|M † (N )± as an element of
Ω
h
%
&
±
×
$± (N )).
+
,
O(Ω)
⊗D
(Z
)
= H 0 (CΩ,h (N ) × X ± , M
HomO(Ω) MΩ† (N )±
p
h
$± (N ) on C (N ) × X ± . We
These glue together into global sections L± (N ) of the sheaf M
$± given by pullback of L± (N ) under i × id, and
finally define L± as the global sections of M
N
then define
L := L+ + L−
(with the obvious meaning).
$N )
∈ H 0 (YN , M
$+ ) ⊕ H 0 (Y − , M
$− )
= H 0 (YN+ , M
N
N
N
6